The foundations of calculus were developed concurrently by Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Because Leibniz was more thoughtful in his notation, his style is more consistent with the notation used today.
Both Newton and Leibniz built the foundations of calculus from the techniques used by mathematicians in calculating infinite processes. Each considered those foundations differently, but all of their standardized techniques came under the derivative and integral. Both men's use of infinitesimals resulted in a reformation of their foundations since infinitesimals are not real. These concepts were later reformulated into limits, giving calculus a more logical foundation.